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Weil pairing and Galois descent

There is an interesting way in which Weil pairing on Abelian Varieties is nothing more than Galois descent for a particular Galois extension. I have not seen this connection used before in the literature but I have also not seen a lot of literature…

Let be an abelian variety over a field of characteristic . Suppose is an integer coprime to and let denote the multiplication by map with kernel .

Recall that there is a dual abelian variety representing the Picard functor for . In particular, is the group of line bundles on such that is trivial. and are finite abelian groups and the Weil pairing is a perfect pairing of the form:

It’s definition goes as follows: Let be a divisor corresponding to and let be a rational function on such that . Then, for , we define .

If denotes translation by , then is the pull back of along . Since , and have the same divisor and so is regular everywhere on and hence constant by properness of . A little more work shows that this rational functions is in . We will see below that this is an immediate conclusion of our alternate viewpoint.

This is the standard picture. However, there is also this alternate way of looking at things:

Consider the etale Galois extension . It’s galois group is canonically identified with . Moreover, is precisely the set of line bundles on trivialized by this etale cover. In other words, it is the set of Galois twists of (or even for any line bundle). Since descent is effective for this extension, this will immediately imply that:

The final equality is because acts trivially on . We also see immediately that since is m-torison as an Abelian group. Therefore, this gives us a pairing:

Explicitly, the map goes as follows: Given a line bundle , we pick an isomorphism . Then, the corresponding 1-cocycle for is defined by . It is easily seen that this is the same explicit construction as in the standard viewpoint.

Conclusion:

I find the Galois descent viewpoint conceptually satisfying. The standard treatments of the Weil pairing can seem arbitrary and it not clear why such a pairing should exist or be useful. On the other hand, is a perfectly natural Galois extension connected to and torsors are clearly an interesting thing to consider.